Multi-Unit Pay-Your-Bid Auction with One-Dimensional Multi-Unit Demands

When the demand functions depend on a one-dimensional private parameter, the two-unit pay-your-bid auction with two homogeneous bidders is equivalent to the one-unit first price auction with two heterogeneous bidders. The existence and uniqueness of the equilibrium of the pay-your-bid auction, its implicit characterization, the «more aggressive» bidding for the second unit, results of comparative statics, and comparisons with the Vickrey auction can be obtained from properties of the first price auction and comparisons with the second price auction.